Probability 1

14 Grade 10 Core Mathematics questions on Probability 1, with worked marking schemes. 14 questions are available on this topic in full.

1. Question 1

   4 marks

   Distinguish between mutually exclusive events and independent events, giving one example of each.

   (4 marks)

   Marking scheme

   Mutually exclusive events: two events that cannot occur at the same time in one trial, so the occurrence of one rules out the other (1 mark). Example: on one roll of a die, obtaining a 2 and obtaining a 5 (1 mark). Independent events: two events in which the occurrence of one does not change the probability of the other (1 mark). Example: tossing a coin and rolling a die, since the outcome on the coin does not affect the die (accept: drawing a bead and replacing it before the second draw) (1 mark).

2. Question 2

   4 marks

   A basket contains 4 ripe and 6 unripe avocados. Two avocados are picked at random, one after the other, without replacement. Determine the probability that (a) both are ripe, and (b) exactly one of them is ripe.

   (4 marks)

   Marking scheme

   (a) P(first ripe) = $\frac{4}{10}$ and P(second ripe given first ripe) = $\frac{3}{9}$ (1 mark). P(both ripe) = $\frac{4}{10}x$ $\frac{3}{9}=\frac{12}{90}=\frac{2}{15}$ (1 mark). (b) Two ways: ripe then unripe = $\frac{4}{10}x$ $\frac{6}{9}=\frac{24}{90}$; unripe then ripe = $\frac{6}{10}x$ $\frac{4}{9}=\frac{24}{90}$ (1 mark for both branches). P(exactly one ripe) = $\frac{24}{90}+\frac{24}{90}=\frac{48}{90}=\frac{8}{15}$ (1 mark).

3. Question 3

   1 mark

   A bag contains 6 red beads and 4 green beads. A bead is picked at random, its colour recorded, and it is returned to the bag before a second bead is picked. What is the probability that both beads picked are green?

   A. $\frac{4}{25}$

   B. $\frac{2}{5}$

   C. $\frac{2}{15}$

   D. $\frac{4}{5}$

   (1 mark)

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4. Question 4

   1 mark

   A fair spinner has eight equal sections numbered 1 to 8. Determine the probability that one spin lands on a multiple of 3 or a multiple of 4.

   A. $\frac{1}{4}$

   B. $\frac{3}{8}$

   C. $\frac{1}{16}$

   D. $\frac{1}{2}$

   (1 mark)

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5. Question 5

   5 marks

   A fair die with faces numbered 1 to 6 is rolled once.

   (a) List the probability space.

   (b) Determine the probability of obtaining a number greater than 4.

   (c) Determine the probability of obtaining a prime number.

   (d) State the probability of obtaining a 7 and give a reason for your answer.

   (5 marks)

   Marking scheme

   (a) S = {1, 2, 3, 4, 5, 6} (1 mark). (b) Favourable outcomes are 5 and 6, so $P=\frac{2}{6}=\frac{1}{3}$ (1 mark). (c) Prime numbers on the die are 2, 3 and 5 (1 mark); $P=\frac{3}{6}=\frac{1}{2}$ (1 mark). (d) P(7) = 0, because 7 is not a member of the probability space, so the event is impossible (1 mark).

6. Question 6

   3 marks

   Records kept by a matatu sacco show that the probability that a particular vehicle arrives late at the stage is 0.15.

   (a) Determine the probability that a trip is not late.

   (b) Out of 60 trips, determine the number of trips expected to be late and the number expected to be on time.

   (3 marks)

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7. Question 7

   2 marks

   A box in a Grade 10 classroom contains 7 blue, 5 red and 8 yellow sticks of chalk. One stick is drawn at random. Determine the probability that it is red, and the probability that it is not yellow.

   (2 marks)

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8. Question 8

   3 marks

   A Grade 10 class tossed a drawing pin 150 times and it landed point-up 96 times.

   (a) State the experimental probability that the pin lands point-up.

   (b) State the experimental probability that it lands point-down.

   (c) Explain why a theoretical probability of 0.5 would not be appropriate for this experiment.

   (3 marks)

   Marking scheme

   (a) P(point-up) = $\frac{96}{150}=\frac{16}{25}=0.64$ (1 mark). (b) P(point-down) = $(150-96)/150=\frac{54}{150}=\frac{9}{25}=0.36$ (1 mark; accept 1 - 0.64). (c) The two outcomes are not equally likely, because the shape and mass of the pin favour one resting position; equally likely outcomes are what a theoretical probability of 0.5 assumes, so here the probability must be estimated from the trials (1 mark).

9. Question 9

   1 mark

   A pencil case holds 5 red pens and 3 blue pens. Two pens are taken out one after the other without replacement. What is the probability that both pens are red?

   A. $\frac{5}{8}$

   B. $\frac{5}{14}$

   C. $\frac{5}{16}$

   D. $\frac{25}{64}$

   (1 mark)

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10. Question 10

    1 mark

    A basket at a fruit stall contains 5 mangoes, 3 avocados and 4 oranges. One fruit is picked at random. What is the probability that the fruit picked is not an orange?

    A. $\frac{1}{3}$

    B. $\frac{1}{2}$

    C. $\frac{3}{4}$

    D. $\frac{2}{3}$

    (1 mark)

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11. Question 11

    1 mark

    A fair coin is tossed once and a fair die is rolled once. What is the probability of obtaining a head on the coin and a number greater than 4 on the die?

    A. $\frac{1}{3}$

    B. $\frac{1}{6}$

    C. $\frac{1}{12}$

    D. $\frac{5}{6}$

    (1 mark)

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12. Question 12

    1 mark

    A Grade 10 learner tossed a bottle top 200 times and it landed open-side up 130 times. What is the experimental probability that this bottle top lands open-side up?

    A. 1.3

    B. 0.35

    C. 0.5

    D. 0.65

    (1 mark)

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13. Question 13

    1 mark

    Which of the following values cannot be the probability of an event?

    A. 0.999

    B. 0

    C. $\frac{3}{8}$

    D. $\frac{7}{5}$

    (1 mark)

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14. Question 14

    3 marks

    The probability that a maize seed from a certain batch germinates is 0.9. Two seeds from the batch are planted, and whether one germinates does not affect the other. Determine the probability that (a) both seeds germinate, (b) neither seed germinates, and (c) at least one seed germinates.

    (3 marks)

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